In a conventional BWR, the core of nuclear fuel is cooled by water. Feedwater is admitted into a reactor pressure vessel (RPV) via a feedwater inlet and a feedwater sparger, which is a ring-shaped pipe having suitable apertures for circumferentially distributing the feedwater inside the RPV. The feedwater flows downwardly through the downcomer annulus, which is an annular region between the RPV 10 and the core shroud 18, which is a stainless steel cylinder surrounding the nuclear fuel core. Water from the downcomer annulus flows radially inwardly to the core lower plenum and then upwardly through the fuel core. A mixture of water and steam flows through standpipes and enters steam separators. The separated liquid water then mixes with feedwater, which mixture then returns to the core via the downcomer annulus.
The BWR also includes a coolant recirculation system which provides the forced convection flow through the core necessary to attain the required power density. A portion of the water is sucked from the lower end of the downcomer annulus via a recirculation water outlet and forced by a centrifugal recirculation pump into jet pump assemblies via recirculation water inlets. The BWR has two recirculation pumps, each of which provides the driving flow for a plurality of jet pump assemblies.
A conventional jet pump assembly comprises an inlet mixer 10 (see FIG. 1A) which receives pressurized driving water via an inlet riser (not shown) and an elbow 14 in flow sequence. A typical BWR has 16 to 24 inlet mixers. In flow sequence starting from the outlet of elbow 12, the inlet mixer comprises: a pre-nozzle section 14; a nozzle section including five nozzles 16 circumferentially distributed at equal angles about the inlet mixer axis; a throat section 18; a barrel section 20; a flare section 22; and a slip joint 24. Each nozzle 16 is tapered radially inwardly at its outlet, so that the nozzle has a maximum diameter d.sub.1 and an exit diameter d.sub.2 which is less than d.sub.1 (see FIG. 1B). The jet pump is energized by this convergent nozzle.
Five secondary inlet openings 26 are circumferentially distributed at equal intervals about the inlet mixer axis. These secondary inlet openings are situated radially outside of the nozzle exits. Therefore, as jets of water exit the nozzles 16, water from the downcomer annulus (not shown) is drawn into the inlet mixer 10 via the secondary inlet openings 26, where it is mixed with water from the recirculation pump (not shown).
Experience has shown that a slow build-up of a "crud" layer on the inner surface of nozzles occurs which gradually degrades their performance. Eventually, this degradation requires that the nozzles be mechanically cleaned, or in extreme cases replaced, which is an expensive and time-consuming operation during plant refueling outages. The build-up of "crud" in jet pumps has been observed in regions of highest flow velocity, caused by the induced electrostatic charge on the metallic inner surface interacting with charged particles suspended in the water. Heaviest deposits have been reported on the inner surfaces of the jet pump throat and nozzle, whereas few, if any, deposits are observed on the diffuser and elbow surfaces. If suspended particulates acquire both positive and negative electrical charges, as they are transported about the BWR, then an electrostatic interaction with the metallic surface can occur. The electrostatic field very close to the surface enhances deposition of particulates carrying charge opposite in sign to the surface charge induced in the metal. This, of course, is also a function of the turbulent bulk-fluid flow and the fluid boundary layer very close to the surface (the laminar sublayer). The fact that deposits build up preferentially in regions of highest bulk-flow velocity can be shown to be a result of higher flows possessing thinner boundary layers, which also effects enhanced deposition.
An extensive analysis of this phenomenon has been performed to elucidate the mechanisms controlling electrostatic deposition of charged particulates in the presence of a laminar fluid sublayer. The idealized physical picture is shown in FIG. 2, where V(x) is the fluid velocity boundary layer and .phi.(y) is the electrostatic boundary layer. An approximate solution of the two-dimensional Navier-Stokes equations for the boundary layer in a convergent nozzle has been obtained to describe the fluid field. The electrostatic field equation is formulated assuming the presence of a volumetric charge density in the fluid, due to the presence of positive- and negative-carriers (or particulates of differing properties and charges). These equations are coupled through the continuity equations for the carrier densities. The resulting set is a ninth-order, nonlinear, two-dimensional set of simultaneous partial differential equations and their boundary conditions. Steady-state conditions are assumed in the approximate solution of this set of equations.
It is found that the electrostatic field possesses a characteristic length .lambda..sub.D which is inversely proportional to the square root of the free-stream particulate density. The fluid field characteristic length is the mean boundary layer thickness .delta. which is inversely proportional to the square root of the Reynolds number Re. The charged-particle density scales as the ratio of these characteristic lengths .delta./.lambda..sub.D, which is proportional to the reciprocal of the product Re.sup.1/4 .lambda..sub.D.
It happens that the character of the solution for the surface flux .GAMMA..sub.-- changes when .delta./.lambda..sub.D .apprxeq.2, manifested as a threshold for electrostatically enhanced particulate deposition. This means that as the flow increases in convergent nozzles, .delta./.lambda..sub.D decreases to the point of deposition threshold. Thereafter, the surface flux is enhanced by the electrostatic forces, resulting in more particulates of higher energy impinging the surface. The resulting "crud layer" on the nozzle surface is tightly adhered, although some erosion by fluid friction can remove the more loosely bound outer portions.
A detailed analysis of the many variables and their interrelations leads to a means of mitigating electro-deposition, so that a nozzle can be designed to operate outside the regime of strong deposition. In order to better understand the instant invention, it is well to summarize the main points of the complicated, lengthy analysis.
A mathematical model of the nozzle depicted in FIG. 2 allows the actual geometry to be reduced to an equivalent one, in which the rate-of-change of velocity with distance along the surface is constant, thereby affording considerable analytical simplification. The free-stream fluid mechanics is described by the Bernoulli equation, in the form: ##EQU1## This pressure drop is due to acceleration alone, neglecting the relatively small inlet and outlet losses and viscous dissipation in the bulk fluid. The assumption is that this pressure drop is the same everywhere in the fluid, including the boundary layer close to the nozzle surface. In addition, the fluid density .rho. is taken to be constant throughout. The bulk fluid velocity V(x) is essentially one-dimensional, but the fluid motion in the boundary layer is two-dimensional. Since the boundary layer is thin compared to the nozzle dimensions, curvature effects are negligible. Using the Bernoulli equation for axial pressure drop and neglecting vertical viscous forces, the continuity and momentum equations in x,y coordinates become: ##EQU2## respectively, where v.sub.x and v.sub.y are the components of the velocity field in the boundary layer defining the stream-function .psi. as: ##EQU3## and applying the similarity transformation: ##EQU4## where the lineal Reynolds number Re.sub.x is defined with respect to free-stream velocity and distance x along the nozzle. Since the flow in the nozzle accelerates with approximately linear axial dependence and the axial rate-of-change of velocity is assumed constant, the pressure gradient is proportional to V(x), which is approximately linear.
The velocity components are given in terms of the similarity variable as: ##EQU5## where the parameter .gamma. is nearly constant: ##EQU6## The bulk velocity increases with x, so the ratio x/V(x) is slowly varying over most of the nozzle length beyond the inlet zone--an important consequence of the simplified nozzle model.
The continuity equation is satisfied identically by the similarity transformation, and after some algebraic manipulation, the momentum equation becomes: ##EQU7## (The primes indicate differentiation with respect to .eta..) This equation is a generalization of the Blasius equation for uniform flow over a flat plate with non-zero pressure gradient (i.e., y=1). Strictly speaking, the accelerated flow field does not conform to a similarity law, but in slowly convergent nozzles, this pseudo-similarity transformation can be applied.
The boundary conditions on v.sub.x and v.sub.y are: ##EQU8## which transform to: EQU f(O)=f'(O)=0; f'(.varies.)=1
The solution for the boundary layer is computed by numerically integrating the nonlinear differential equation for f(.eta.). The result is then used to compute the velocity distribution in the boundary layer. The tangential component is much larger than the normal component of velocity, therefore only the tangential velocity distribution need be considered further. This is shown in FIG. 3 for a typical case.
The boundary layer narrows as the flow accelerates in the convergent section of the nozzle, as seen from the shape of the fluid velocity distribution. The boundary layer thickness .delta.0 is defined to be the value of y corresponding to 99% of the free-stream velocity at any particular tangential position (x-value). .delta.(x) is computed iteratively from the velocity field solution, with the result shown in FIG. 4 for a typical case (L=11.8 in.). The mean value of the curve in FIG. 4 is what enters into the subsequent description.
In the free stream of the fluid there are suspended at least two species of charged particles one carrying charge q.sub.+ and the other carrying charge q.sub.-, with respective densities n.sub.+ and n.sub.-. The various species are in thermal equilibrium with the fluid; therefore, each possesses a Maxwellian velocity distribution that corresponds to the fluid temperature T (.degree.K.). If the dielectric constant of the fluid is K, the Gauss law for the electrostatic field intensity E, or the reduced potential .PHI., can be written as: ##EQU9## where .sub.0 (8.855.times.10.sup.-12 farad/m) is the permittivity of vacuum and k (0.8617.times.10.sup.-4 eV/.degree.K.) is the Boltzmann constant. Here, the following definitions are used: ##EQU10## The charged-particle densities are each functions of position, but the charges they carry are assumed to be constant (radiation field effects are negligible in jet pump locations). Each specie possesses a mobility in the fluid (.mu..sub.+ or .mu..sub.-), which are numerically different.
The variation of n.sub.0 with x is due to convergence of the flow that carries the species in the bulk fluid. Continuity requires: ##EQU11## where N.sub.0 V.sub.0 is the free-stream inlet flux density. Then defining the pseudo-constant .chi. by: ##EQU12## and transforming to the .eta.-variable yields: ##EQU13## The boundary conditions on .PHI. are: ##EQU14## which transform to: ##EQU15## The quantity .sigma. is the induced surface charge density, and Re.sub.L is defined as Re.sub.x [L].
The nonlinear differential equation for .PHI.(.eta.) is inhomogeneous, with n.sub.-- (.eta.) acting like a driving function. To determine n.sub.-- (.eta.), we apply a continuity equation to the particle flux .GAMMA..sub.-- of negative-carriers, and perform some mathematical manipulations to obtain: ##EQU16## Since f(.eta.) is known for the nozzle boundary layer, the coupled .zeta.-equation and .PHI.-equation can be solved numerically and their boundary conditions satisfied. Then, the surface flux .GAMMA..sub.S can be calculated from these numerical solutions. The results are shown in FIGS. 5-7.
The symbol in FIG. 7 denotes a typical case for jet pump nozzles. This function exhibits a threshold at .delta./.lambda..sub.D .apprxeq.2.15, implying that there exists a critical value of the mean boundary layer thickness for which strong deposition can occur.
The normalized electric field, due to the space-charge distribution near the metallic surface, is shown in FIG. 8 for a typical case involving enhanced mass transfer. Although the surface potential is typically a few tens of millivolts, the electric field intensity at the surface E.sub.0 can be quite large, since the length scale is on the order of .lambda..sub.D, a very small distance (.about.1 .mu.in.) on the order of the fluid boundary layer thickness.